Vlasov-Maxwell equation: Difference between revisions

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The '''Vlasov-Maxwell equations''' equations are given by  
The '''Vlasov-Maxwell equations''' equations are given by  


:<math>f_t + \underline{v} \cdot \nabla_x f + (E + \underline{v} \times B) \cdot \nabla_{\underline{v}} f = 0 </math>
:<math>f_t + \underline{v} \cdot \nabla_x f + (E + \underline{v} \times B) \cdot \nabla_v f = 0 </math>
:<math>\nabla \cdot E = 4 \pi \rho </math>
:<math>\nabla \cdot E = 4 \pi \rho </math>
:<math>\nabla \cdot B = 0 </math>
:<math>\nabla \cdot B = 0 </math>
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:<math>B_t = - \nabla \times E</math>
:<math>B_t = - \nabla \times E</math>


where <math>f(t,x,\underline{v})</math> is the particle density (and is non-negative), <math>j(t,x) = \int \underline{v} f(t,x,\underline{v}) dv</math> is the current density, <math>\rho(t,x) = \int f(t,x,\underline{v}) dv</math> is the charge density, and <math>\underline{v} = v / (1 + |v|^2)^{1/2}</math> is the relativistic velocity.  The vector fields E(t,x) and B(t,x) represent the electromagnetic field.  x and v live in R^3 and t lives in R.  This equation is a coupled wave and conservation law system, and models collision-less plasma at relativistic velocities.  
where <math>f(t,x,v)</math> is the particle density (and is non-negative), <math>j(t,x) = \int \underline{v} f(t,x,v) dv</math> is the current density, <math>\rho(t,x) = \int f(t,x,v) dv</math> is the charge density, and <math>\underline{v} = v / (1 + |v|^2)^{1/2}</math> is the relativistic velocity.  The vector fields E(t,x) and B(t,x) represent the electromagnetic field.  x and v live in R^3 and t lives in R.  This equation is a coupled wave and conservation law system, and models collision-less plasma at relativistic velocities.  
    
    
Assuming that the particle density remains compactly supported in the velocity domain for all time, GWP in C^1 was proven in [[GsSr1986b]] (see also [[GsSr1986]], [[GsSr1987]].  
Assuming that the particle density remains compactly supported in the velocity domain for all time, GWP in C^1 was proven in [[GsSr1986b]] (see also [[GsSr1986]], [[GsSr1987]].  

Latest revision as of 13:33, 11 July 2007

The Vlasov-Maxwell equations equations are given by

where is the particle density (and is non-negative), is the current density, is the charge density, and is the relativistic velocity. The vector fields E(t,x) and B(t,x) represent the electromagnetic field. x and v live in R^3 and t lives in R. This equation is a coupled wave and conservation law system, and models collision-less plasma at relativistic velocities.

Assuming that the particle density remains compactly supported in the velocity domain for all time, GWP in C^1 was proven in GsSr1986b (see also GsSr1986, GsSr1987. An alternate proof of this result is in KlSt2002. A stronger result (which only imposes compact support conditions on the initial data, not on all time) regarding solutions to Vlasov-Maxwell which are purely outgoing (no incoming radiation) is in Cal-p. The velocity demain hypothesis can be removed in the "2.5 dimensional model" where the x_3 dependence is trivial but the v_3 dependence is not GsScf1990. Further results are in GsSch1988, Rei1990, Wol1984, Scf1986 The non-relativistic limit of Vlasov-Maxwell is Vlasov-Poisson, in which the electromagnetic field is replaced by . Considerably more is known for the existence theory of this equation.